EXAMPLE 1
Use the Law of Detachment
Use the Law of Detachment to make a valid
conclusion in the true situation.
a. If two segments have the same length, then they are
congruent. You know that BC = XY.
b. Mary goes to the movies every Friday and Saturday
night. Today is Friday.
SOLUTION
a. Because BC = XY satisfies the hypothesis of a true
conditional statement, the conclusion is also true.
So, BC = XY.
EXAMPLE 1
Use the Law of Detachment
b. First, identify the hypothesis and the conclusion of
the first statement.The hypothesis is “If it is Friday
or Saturday night,” and the conclusion is “then
Mary goes to the movies.”
“Today is Friday” satisfies the hypothesis of the
conditional statement, so you can conclude that
Mary will go to the movies tonight.
EXAMPLE 2
Use the Law of Syllogism
If possible, use the Law of Syllogism to write a new
conditional statement that follows from the pair of true
statements.
a. If Rick takes chemistry this year, then Jesse will be
Rick’s lab partner.If Jesse is Rick’s lab partner, then
Rick will get an A in chemistry.
b. If x2 > 25, then x2 > 20.
If x > 5, then x2 > 25.
c. If a polygon is regular, then all angles in the interior
of the polygon are congruent.
If a polygon is regular, then all of its sides are
congruent.
EXAMPLE 2
Use the Law of Syllogism
SOLUTION
a. The conclusion of the first statement is the
hypothesis of the second statement, so you can
write the following new statement.
If Rick takes chemistry this year, then Rick will get
an A in chemistry.
b. Notice that the conclusion of the second statement
is the hypothesis of the first statement, so you can
write the following new statement.
If x > 5, then x2 > 20.
EXAMPLE 2
Use the Law of Syllogism
c. Neither statement’s conclusion is the same as the
other statement’s hypothesis. You cannot use the
Law of Syllogism to write a new conditional
statement.
GUIDED PRACTICE
for Examples 1 and 2
1. If 90° < m R < 180°, then R is
obtuse. The measure of R is 155°.
Using the Law of Detachment, what
statement can you make?
ANSWER
R is obtuse
GUIDED PRACTICE
for Examples 1 and 2
2. If Jenelle gets a job, then she can afford a car. If
Jenelle can afford a car, then she will drive to
school. Using the Law of Syllogism, what
statement can you make ?
ANSWER
Notice that the conclusion of the first statement is
the hypothesis of the second statement, So you
can write the following statement.
If Jenelle gets a job, then she will drive to school.
GUIDED PRACTICE
for Examples 1 and 2
State the law of logic that is illustrated.
3. If you get an A or better on your math test, then
you can go to the movies. If you go to the
movies, then you can watch your favorite actor.
If you get an A or better on your math test, then
you can watch your favorite actor.
ANSWER
Law of Syllogism.
GUIDED PRACTICE
for Examples 1 and 2
4. If x > 12, then x + 9 > 20. The value of x is 14
Therefore, x + 9 > 20
ANSWER
Law of Detachment
EXAMPLE 3
Use inductive and deductive reasoning
ALGEBRA What conclusion can you make about
the product of an even integer and any other
integer?
SOLUTION
STEP 1
Look: for a pattern in several examples. Use inductive
reasoning to make a conjecture.
(–2) (2) = –4, (–1) (2)= –2, 2 (2)= 4, 3 (2)= 6,
(–2) (–4) = 8, (–1) (–4) = 4, 2 (–4)= –8, 3 (–4) = –12
Conjecture: Even integer
Any integer = Even integer
EXAMPLE 3
Use inductive and deductive reasoning
STEP 2
Let: n and m each be any integer. Use deductive
reasoning to show the conjecture is true.
2n is an even integer because any integer multiplied by 2
is even.
2nm represents the product of an even integer
and any integer m.
2nm is the product of 2 and an integer nm. So, 2nm is
an even integer.
ANSWER
The product of an even integer and any integer is
an even integer.
EXAMPLE 4
Reasoning from a graph
Tell whether the statement is
the result of inductive
reasoning or deductive
reasoning.
Explain your choice.
a. The northern elephant
seal requires more
strokes to surface the
deeper it dives.
b. The northern elephant seal uses more strokes
to surface from 250 meters than from 60 meters.
EXAMPLE 4
Reasoning from a graph
SOLUTION
a. Inductive reasoning, because it is based on a
pattern in the data
b. Deductive reasoning, because you are comparing
values that are given on the graph
GUIDED PRACTICE
for Examples 3 and 4
5. Use inductive reasoning to make a conjecture
about the sum of a number and itself. Then use
deductive reasoning to show the conjecture is true.
SOLUTION
Conjecture: The sum of a number and itself is twice
the number.
Deductive reasoning: Let n be any integer. Use
deductive reasoning to show the conjecture is true
n + n = 2n 
GUIDED PRACTICE
for Examples 3 and 4
6. Use inductive reasoning to write another statement
about the graph in Example 4. Then use deductive
reasoning to write another statement.
SOLUTION
Using inductive reasoning: The more strokes it takes
for the northern elephant to surface, the deeper it
dove.
Using deductive reasoning: The northern elephant
seal uses fewer strokes to surface from 190 meters
then from 410 meters.